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The Strict Topology in a Completely Regular Setting: Relations to Topological Measure Theory

Published online by Cambridge University Press:  20 November 2018

Steven E. Mosiman  and
Robert F. Wheeler
Steven E. Mosiman
Affiliation:
University of Missouri, Columbia, Missouri
Robert F. Wheeler
Affiliation:
Louisiana State University, Baton Rouge, Louisiana
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Let X be a locally compact Hausdorff space, and let C*(X) denote the space of real-valued bounded continuous functions on X. An interesting and important property of the strict topology β on C*(X) was proved by Buck [2]: the dual space of (C*(X), β) has a natural representation as the space of bounded regular Borel measures on X.

Now suppose that X is completely regular (all topological spaces are assumed to be Hausdorff in this paper). Again it seems natural to seek locally convex topologies on the space C*(X) whose dual spaces are (via the integration pairing) significant classes of measures.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 5 , October 1972 , pp. 873 - 890
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Arhangel'skii, A. V., Bicompact sets and the topology of spaces, Trans. Moscow Math. Soc. 13 (1965), 1–62.Google Scholar
2. Buck, R. C., Bounded continuous functions on a locally compact space, Michigan Math. J. 5 (1958), 95–104.Google Scholar
3. Choquet, G., Sur les ensembles uniformément négligeables, Sém. Choquet (1969/70), No. 6.Google Scholar
4. Conway, J. B., The strict topology and compactness in the space of measures, Trans. Amer. Math. Soc. 126 (1967), 474–486.Google Scholar
5. Dudley, R. M., Convergence of Baire measures, Studia Math. 27 (1966), 251–268.Google Scholar
6. Dugundji, J., Topology (Allyn and Bacon, Boston, 1966).Google Scholar
7. Fernique, X., Processus linéaires, processus généralises, Ann. Inst. Fourier (Grenoble) 17 (1967), 1–92.Google Scholar
8. Fremlin, D. H., Garling, D. J. H., and Haydon, R. G., On measures on topological spaces, Proc. Internat. Colloquium, Bordeaux, 1971 (to appear).Google Scholar
9. Giles, R., A generalization of the strict topology, Trans. Amer. Math. Soc. 161 (1971), 467–474.Google Scholar
10. Gillman, L. and Jerison, M., Rings of Continuous Functions (Van Nostrand, Princeton, 1960).Google Scholar
11. Hoffmann-Jørgensen, J., A generalization of the strict topology, Aarhus Universitet Preprint Series 1969/70, No.Google Scholar
12. Kirk, R. B., Locally compact, B-compact spaces, Nederl. Akad. Wetensch. Proc. Ser. A. 72 (1969), 333–344. 32.Google Scholar
13. Kirk, R. B., Measures in topological spaces and B-compactness, Nederl. Akad. Wetensch. Proc. Ser. A. 72 (1969), 172–183.Google Scholar
14. Knowles, J. D., Measures on topological spaces, Proc. London Math. Soc. 17 (1967), 139–156.Google Scholar
15. LeCam, L., Convergence in distribution of stochastic processes, Univ. of California, Publ. in Statistics 2 (1953-8), 207-236.Google Scholar
16. Michael, E. A., Local compactness and cartesian products of quotient maps and k-spaces, Ann. Inst. Fourier (Grenoble) 18 (1968), 281–286.Google Scholar
17. Moran, W., The additivity of measures on completely regular spaces, J. London Math. Soc. 43 (1968), 633–639.Google Scholar
18. Moran, W., Measures and mappings on topological spaces, Proc. London Math. Soc. 19 (1969), 493–508.Google Scholar
19. Moran, W., Measures on metacompact spaces, Proc. London Math. Soc. 20 (1970), 507–524.Google Scholar
20. Negrepontis, S., Baire sets in topological spaces, Arch. Math. (Basel) 18 (1969), 603–608.Google Scholar
21. Noble, N., k-spaces and some generalizations, Doctoral Dissertation, University of Rochester, 1967.Google Scholar
22. Preiss, D., Metric spaces in which Prohorov's theorem is not valid, Proc. Prague Symposium on General Topology, 1971 (to appear).Google Scholar
23. Schaefer, H. H., Topological vector spaces (Macmillan, New York, 1966).Google Scholar
24. Sentilles, F. D., Bounded continuous functions on a completely regular space (to appear).Google Scholar
25. Summers, W. H., The general complex bounded case of the strict weighted approximation property, Math. Ann. 192 (1971), 90–98.Google Scholar
26. Taylor, D. C., A general Phillips Theorem for C*-algebras and some applications, Pacific J. Math. 40 (1972), 477–488.Google Scholar
27. Topsøe, F., Compactness in spaces of measures, Studia Math. 36 (1970), 195–212.Google Scholar
28. Topsøe, F., Topology and measure (Springer-Verlag lecture notes, vol. 133, 1970).Google Scholar
29. Van Rooj, A. C. M., Tight Junctionals and the strict topology, Kyungpook Math. J. 7 (1967), 41–43.Google Scholar
30. Varadarajan, V. S., Measures on topological spaces, Amer. Math. Soc. Transl. 48 (1965), 161–228.Google Scholar
31. Warner, S., The topology of compact convergence on continuous function spaces, Duke Math. J. 25 (1958), 265–282.Google Scholar
32. Willard, S., General topology (Addison-Wesley, Reading, Mass., 1970).Google Scholar